We solve the problem of hedging contingent claim in an incomplete market under model uncertainty. The quality of a hedging strategy is measured by the robust exponential utility from the net terminal wealth. The uncertainty is expressed as a convex set of probability measures. Our method is based on the convex duality theory. The dual problem involves the minimization of a penalized robust f-divergence over the set of martingale measures. We first show the existence and uniqueness of the solution to the dual problem, which is a slight extension of the result by Follmer and Gundel (2006, Illinois J. Math.). Then a representation of the solution will be given in the form of a martingale representation with respect to the price process of the underlying assets. Finally, we show that the solution to the original hedging problem is expressed by the solution to the dual problem. At the end of this paper, an explicitly solvable example is presented in the setting of a stochastic factor model for which we can solve the dual problem by the stochastic control technique. We derive the corresponding HJB equation and the representation of the optimal strategy by the value function.